In the first part devoted to logREMs, we show how to characterize their common properties and model--specific data. Then we develop their replica symmetry breaking treatment, which leads to the freezing scenario of their free energy distribution and the general description of their minima process, in terms of decorated Poisson point process. We also report a series of new applications of the Jack polynomials in the exact predictions of some observables in the circular model and its variants. Finally, we present the recent progress on the exact connection between logREMs and the Liouville conformal field theory.

The goal of the second part is to introduce and study a new class of banded random matrices, the broadly distributed class, which is characterid an effective sparseness. We will first study a specific model of the class, the Beta Banded random matrices, inspired by an exact mapping to a recently studied statistical model of long--range first--passage percolation/epidemics dynamics. Using analytical arguments based on the mapping and numerics, we show the existence of localization transitions with mobility edges in the ``stretch--exponential'' parameter--regime of the statistical models. Then, using a block--diagonalization renormalization approach, we argue that such localization transitions occur generically in the broadly distributed class.

The defense presentation will focus on the logREM--Liouville connection and the broadly distributed random matrices.

Aleksey FEDOROV

28 juin 2017

Auditorium Irène Joliot-Curie de l'IPN

Soutenance de thèse

Non-conventional many-body phases in ultracold dipolar systems

The problem of revealing and describing novel macroscopic quantum states characterized by exotic and non-conventional properties has the fundamental importance for modern physics. Such states offer fascinating prospects for potential applications in quantum information processing, quantum simulation, and material research. In the present Thesis we develop a theory for describing non-conventional phases of ultracold dipolar gases. The related systems of large-spin atoms, polar molecules, and dipolar excitons in semiconductors are actively studied in experiments. We put the main emphasis on revealing the role of the long-range character of the dipole-dipole interaction. We consider the effect of rotonization for a 2D weakly interacting gas of tilted dipolar bosons in a homogeneous layer, and demonstrate that in contrast to the case of perpendicular dipoles, in a wide range of tilting angles the condensate depletion remains small even when the roton minimum is extremely close to zero. We predict the effect of rotonization for a weakly correlated Bose gas of dipolar excitons in a semiconductor layer and calculate the stability diagram. According to our estimates, the threshold of the roton instability for a bose-condensed exciton gas with the roton-maxon spectrum is achievable experimentally in semiconductor layers. We then consider p-wave superfluids of identical fermions in 2D lattices. The optical lattice potential manifests itself in an interplay between an increase in the density of states on the Fermi surface and the modification of the fermion-fermion interaction (scat- tering) amplitude. The density of states is enhanced due to an increase of the effective mass of atoms. For short-range interacting atoms in deep lattices the scattering amplitude is strongly reduced compared to free space due to a small overlap of wavefunctions of fermions sitting in the neighboring lattice sites, which suppresses the p-wave superfluidity. However, we show that for a moderate lattice depth there is still a possibility to create p-wave superfluids with sizable transition temperatures. For fermionic polar molecules, due to a long-range character of the dipole-dipole interaction the effect of the suppression of the scattering amplitude is absent. It is shown that for microwave-dressed polar molecules a stable topological p+ip superfluid may emerge in the 2D lattice at realistic temperatures. Finally, we discuss another interesting novel superfluid of fermionic polar molecules. It is expected in a bilayer system, where dipoles are oriented perpendicularly to the layers and in opposite directions in different layers. We demonstrate the emergence of interlayer superfluid pairing. In contrast to the already known s-wave interlayer superfluid, when all dipoles are parallel to each other, in our case the s-wave pairing is suppressed and there can be p-wave or higher partial wave superfluids.

Maxime SEVELEV

6 octobre 2017

Auditorium Irène Joliot-Curie de l'IPN

Soutenance de thèse

Phase diagram, jamming and glass transitions in the non-convex perceptron

This thesis treats the «spherical perceptron model», a simple exactly solvable model for glassy behavior and jamming suitably generalized to negative values of scalar product parameter κ. The classical machine-learning problem of random pattern classification by the perceptron is a convex constraint satisfaction problem (CSP). Even when the «stability parameter» κ of the model becomes negative, the problem still makes sense and can be interpreted as the problem of particles on an N-dimensional sphere trying to avoid randomly placed obstacles. In this case, the corresponding CSP is non-convex. This thesis studies the problem in detail in the non-convex domain. Systematic study is made possible by assigning to a constraint satisfaction problem its corresponding optimization version endowed with a Hamiltonian function (cost function) quantifying the violations of the constraints, as a function of the system's configuration. The connection between random CSP and glassy phenomenology in physics is well known and has been explored in detail for models with discrete variables. The presence of continuous variables in the (spherical) perceptron model enables us to unveil, in random CSP, the characteristic SAT/UNSAT transition where the system transits from the satisfiable regime (where the ground state has zero energy) to the unsatisfiable one (where the ground state energy is positive). This phase transition can also be interpreted as a jamming transition similar to the one that exhibit models with frictionless spheres. The simplicity of the considered model allows the exact determination of the zero temperature phase diagram as a function of the control parameters: the density of obstacles and their size. In the present thesis, the jamming transition thus identified is completely characterized and several glass phases of stable and marginal character are studied in detail.

Angelika KNOTHE

10 octobre 2017

Albert-Ludwigs-Universität Freiburg

Soutenance de thèse

Quantum Hall Ferromagnetism in multicomponent

Within the framework of Quantum Hall Ferromagnetism [1, 2] we theoretically investigatethe properties of different two-dimensional electron systems in the quantum Hallregime in which the electrons may be endowed with multiple discrete degrees of freedom.Different orderings of these electronic degrees of freedom, treated as different spins andisospins, characterise different possible phases of the system. Using Hartree Fock meanfield theory we treat several examples: We analyse how the structure of ground andexcited states in a finite piece of monolayer graphene, where the electrons carry besidesthe electronic spin also a valley-isospin degree of freedom, is influenced by the presence ofa boundary with respect to the bulk properties [3]. The properties of the resulting edgestate structure is discussed. For a graphene bilayer, where an additional orbital isospincomes into play due to a degeneracy between the n=0 and the n=1 Landau level, weidentify the various different ground state spin and isospin phases which emerge as functionsof external electric and magnetic fields and investigate the properties of phases thatexhibit non-trivial coherence properties of the isospin degrees of freedom [4]. Finally, asan outlook, possible extensions to two-dimensional surface states of three-dimensionalcrystals are discussed.